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Triangle T(n,k) of elements of n-th Weyl group of type B whose reduced word uses n-k generators.
2

%I #14 Nov 25 2017 11:13:30

%S 1,1,1,5,2,1,35,9,3,1,309,56,14,4,1,3287,443,84,20,5,1,41005,4298,623,

%T 120,27,6,1,588487,49937,5629,859,165,35,7,1,9571125,680700,61300,

%U 7360,1162,220,44,8,1,174230863,10683103,793402,75714,9584,1544,286,54,9,1

%N Triangle T(n,k) of elements of n-th Weyl group of type B whose reduced word uses n-k generators.

%C Row sums are 2^n n!.

%C G.f. for k-th column is given by (1-1/g(x))^(k-1)*g(2x)/g(x).

%H N. Bergeron, C. Hohlweg, M. Zabrocki, <a href="http://arXiv.org/abs/math.CO/0509271">Posets related to the connectivity set of Coxeter groups</a>, arXiv:math/0509271 [math.CO], 2005-2006.

%F G.f.: g(2x)/(t+(1-t)g(x)) where g(x) = sum_{n>=0} n! x^n.

%e T(3,1)=9 because B_3 is generated by {t,s1,s2} where t^2=s1^2=s2^2=(s1 s2)^3=(t s1)^4=(t s2)^2=1.

%e The 9 elements which only use 2 generators are {s1 s2, s1 s2 s1, s2 s1, s2 t, t s1, s1 t s1, s1 t s1 t, s1 t, t s1 t}.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 5, 2, 1;

%e 35, 9, 3, 1;

%e 309, 56, 14, 4, 1;

%e ...

%p f:=proc(n,k) local gx; gx:=add(i!*x^i,i=0..n); coeff(series((1-1/gx)^k*subs(x=2*x,gx)/gx,x,n+1),x,n); end:

%t nmax = 9;

%t g[x_] = Sum[n!*x^n, {n, 0, nmax}];

%t gf[x_, t_] = g[2*x]/(t + (1 - t)*g[x]);

%t T[n_, k_] := SeriesCoefficient[gf[x, t], {x, 0, n}] // SeriesCoefficient[#, {t, 0, k}]&;

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 25 2017 *)

%Y Cf. A109253, A003319, A085771, A059438.

%Y For the similar sequence in type D, see A112226.

%K nonn,tabl

%O 0,4

%A _Mike Zabrocki_, Aug 19 2005